where \(\mathcal {H}_{\mathrm {QAH}}(\mathbf {k})=\mathbf {h}(\mathbf {k})\cdot \)  describes the QAH phase [1]}, [2]}, with Bloch vector \(\mathbf {h}=(\xi _{\text{so}}\sin k_x,\xi _{\text{so}}\sin k_y,m_z-\xi _0\cos k_x-\xi _0\cos k_y)\) . Here \(\xi _{0}\)  (or \(\xi _{\text{so}}\) ) simulates the spin-conserved (spin-flipped) hopping coefficient, and \(m_{z}\)  is the magnetic field. The white noise \(w_{i}(\mathbf {k},t)\)  of strength \(\sqrt{w_{i}}\)  couples to the Pauli matrix \(\sigma _{i}\)  and satisfies \(w_i(\mathbf {k},t) _{\mathrm {noise}} =0\)  and \(w_i(\mathbf {k},t)w_j(\mathbf {k},t^{\prime }) _{\mathrm {noise}} =w_i\delta _{ij}\delta (t-t^{\prime })\) , where \(\cdot _{\mathrm {noise}}\)  is the stochastic average over different noise configurations. Without noise, the Hamiltonian \(\mathcal {H}_{\mathrm {QAH}}\)  hosts nontrivial QAH phase for \(0<|m_z|<|\xi _0|\)  with Chern number \(C_1=\mbox{sgn}(m_z)\) , and the phase is trivial for \(|m_z|>|\xi _0|\)  or \(m_z=0\)  [1]}.
